Curry's Paradox

Curry's paradox, so named for its discoverer, namely Haskell B. Curry,
is a paradox within the family of so-called paradoxes of self-reference
(or paradoxes of circularity). Like the liar paradox (e.g., ‘this
sentence is false’) and
Russell's paradox,
Curry's paradox challenges familiar naive theories,
including naive truth theory (unrestricted T-schema) and naive set
theory (unrestricted axiom of abstraction), respectively. If one
accepts naive truth theory (or naive set theory), then Curry's paradox
becomes a direct challenge to one's theory of logical implication or
entailment. Unlike the liar and Russell paradoxes Curry's paradox is
negation-free; it may be generated irrespective of one's theory of
negation. An intuitive version of the paradox runs as follows.

Consider the following list of sentences, named ‘The
List’.

Tasmanian devils have strong jaws.

The second sentence on The List is circular.

If the third sentence on The List is true, then every
sentence is true.

The List comprises exactly four sentences.

Although The List itself is not paradoxical, the third sentence (a
conditional) is. Is it true? Well, suppose, for conditional proof, that
its antecedent is true. Then

the third sentence of The List is true

is true. Since the third sentence of The List is `If the third
sentence of The List is true, then every sentence is true',
substitution (of identity) yields that

If the third sentence of The List is true, then every
sentence is true

is true. But, then, Modus Ponens on the above two sentences yields
that

every sentence is true

is true. So, by conditional proof, we conclude that

If the third sentence of The List is true, then every
sentence is true

is true. By substitution (of identity, as above), it follows that

the third sentence of The List is true

is true. But, now, by Modus Ponens on the above two sentences we get
that

every sentence is true

is true. By the Release rule for
truth,[1]
we conclude: every sentence is true! So goes (one version of)
Curry's paradox.

In 1942 Haskell B. Curry presented what is now called Curry's
paradox. Perhaps the most intuitive version of the paradox is due
to Arthur N. Prior (1955), who recast Curry's paradox as a ‘proof’ of
God's existence. (Let C = ‘If C is true then
God exists’.) The version presented above is (in effect) Prior's
version.

There are basically two different versions of Curry's paradox, a
truth-theoretic (or proof-theoretic) and a set-theoretic version;
these versions will be presented below. (There is also a third one
concerning properties, which may or may not be the same as the
set-theoretic version; it depends on whether sets and
semantical properties are the same thing. This is briefly
touched on below.) For now, however, there are a few caveats that need
to be issued.

Caveat 1. Löb's Paradox. Prior's version is (in effect)
rehearsed by Boolos and Jeffrey (1989), where neither Prior nor Curry
is given credit; rather, Boolos and Jeffrey point out the similarity of
the paradox to reasoning used within the proof of Löb's Theorem; and
subsequent authors, notably Barwise & Etchemendy (1984), have
called the paradox Löb's paradox. While there is no doubt
strong justification for the alternative name (given the similarity of
Curry's paradox to the reasoning involved in proving Löb's Theorem)
the paradox does appear to have been first discovered by Curry.

Caveat 2. Geometrical Curry Paradox (Jigsaw Paradox). This
is not the same Curry paradox under discussion; it is a
well-known paradox, due to Paul Curry, having to do with
so-called geometrical dissection. (The so-called Banach-Tarski
geometrical paradox is related to Paul Curry's geometrical
paradox.) See Gardner 1956 and Fredrickson 1997 for full discussion of
this (geometrical) Curry paradox.

2.3 Property-Theoretic Version

On the surface, it's plausible that every meaningful predicate
corresponds (or ‘expresses’, or ‘denotes’, or
whathaveyou) a semantical property. In particular, for every
meaningful predicate F, there is a semantical
property <(y : F(y))> such that an object
x exemplifies <(y : F(y))> just if
F is true of x. Letting E be the
exemplification relation, this unrestricted semantical property
principle (USP) takes a familiar form, namely,

USP: xE<(y : F(y))> ↔
F(x).

Curry's property-theoretic paradox results from replacing ∈
(membership) with E (exemplification) in Unrestricted
Abstraction (UA) and replacing the set-notation with the
‘property’
notion.[2]

3.1 Significance

What is the significance of Curry's paradox? The answer depends on
one's approach to paradox in general. Any comprehensive theory of
language has to give some sort account of the paradoxes (e.g., the
liar, or Russell's, or etc.). Classical approaches tend to fiddle with
the T-schema (or naive abstraction) or reject the existence of certain
(paradoxical) sentences. Such classical approaches tend to respond to
Curry's paradox in the same fashion — by rejecting the existence
of Curry sentences or fiddling with the unrestricted T-schema (or
naive abstraction). Some popular variations of these two options
include Gupta-Belnap revision theory (1993), Tarski's familiar
hierarchical theory (or Russellian type theory), Simmons's singularity
theory (1993), Burge's indexical theory (1979), Kripke's fixed point
semantics (1975), Gaifman's pointer semantics (1988),
Barwise-Etchemendy situation-cum-Aczel-set-theory (1984), Glanzberg's
contextual semantics (2001) and, similarly Parsons (1974), and
others. (NB: These theories are quite different from each other;
however, each of them fits under one of the two so-called classical
options mentioned above; they either modify the naive T-schema or
reject the existence of so-called strengthened liar sentences.)

Where Curry's paradox becomes especially significant is not with
classical approaches but rather with certain non-classical
approaches. In particular, Curry's paradox is a direct challenge to
any non-classical approach that attempts to preserve one of the
canvassed schemas — truth, sets, semantical properties — in
unrestricted form (while nonetheless allowing Curry sentences to arise
in the language).

The two most prominent non-classical approaches, at least with respect
to dealing with Curry, are
paraconsistent
and
paracomplete approaches. The former affords (negation-)
inconsistent but non-trivial theories (by rejecting
‘explosion’, the rule from a sentence and its negation to
an arbitrary sentence). The latter rejects ‘excluded
middle’, that is, the validity of the disjunction of a sentence
and its negation.[3]

In the target non-classical theories (all of which have classical
logic as a proper extension), the so-called ‘material
conditional’ (the ‘hook’ or ‘horseshoe’)
is not a terribly adequate conditional, where the hook is a disguised
disjunction, namely ¬A∨B. In target
paracomplete theories, the hook fails to satisfy
‘Identity’ (validity of ‘if A then
A’, in the hook sense of ‘if’). In the
target paraconsistent theories, the hook fails to detach (fails to
satisfy ‘modus ponens’). So, the target non-classical
theories tend to seek out a conditional that goes beyond the hook.

To focus matters a bit more (though still keeping things slightly
loose), let a suitable conditional be a conditional with the
following minimal features.

Identity: A→A is valid.

(Rule) Modus Ponens: The argument (form) from A and A→B to B is valid.

Curry-generating “contraction” is avoided!

There are other constraints one might impose (e.g., substitutivity of
equivalents, avoidance of omega-inconsistency in arithmetic, perhaps
more), but the above conditions serve as a minimal constraint on a
suitable
conditional.[4]

What Curry's paradox teaches us is that, regardless of their merits
with respect to simple Liar-like sentences, not just any old
paraconsistent or paracomplete theory will work if we're to preserve
one or more of the given (unrestricted) principles. On pain of
triviality, no connective in the language can satisfy contraction or
absorption and support the Truth or UA or USP schemes (at least if the
logic is otherwise relatively normal). Among other things, this
constraint rules out quite a few popular candidates for (otherwise
suitable) conditionals — including, for example, various popular
relevant
conditionals, including those of E and
R.

In the next two subsections, I sketch two basic (in various ways
related) approaches to a suitable conditional, so understood. Details
and refinements (of which there are many!) are left to cited work.

3.2 A simple non-normal worlds solution

One family of approaches to a suitable conditional invokes non-normal
worlds (originally introduced by Kripke for purposes of modeling modal
logics for which Necessitation fails). A particularly simple
approach, which suits either paraconsistent or paracomplete (or both)
approaches to paradox, is due to Graham Priest (1992) and, in a
slightly different form, Routley and Loparic (1973). The idea may be
seen easily through its semantics, as
follows.[5]

Setting negation aside (for purposes of Curry), we assume a
propositional language with the following connectives: conjunction
(&), disjunction (∨), and entailment (→). (For purposes of
resolving Curry's paradox, negation may be set aside; however, the
current semantics allow for a variety of approaches to negation, as
well as quantifiers.) An interpretation is a 4-tuple,
(W,N,[ ], f), where W is a
non-empty set of worlds (index points), N is a non-empty
subset of W, [ ] is a function from propositional
parameters to the powerset of W; we may, for convenience, see
the range of [ ] as comprising propositions (sets of worlds at
which various sentences are true), and so call the values of [ ]
propositions. We let NN be the set of so-called
non-normal worlds, namely NN =
W−N. In turn, f is a function from
(ordered) pairs of propositions to NN. Now, [ ] is
extended to all sentences (A, B, …) via the
following clauses:

[A&B] = [A]∩[B]

[A∨B] = [A]∪[B]

The value of an entailment is the union of two sets: N, the
class of normal worlds where the entailment is true, and NN,
the of non-normal worlds where the entailment is true. Assuming the
usual S5 truth conditions, N and NN are
specified thus:

N = W, if [A]⊂[B]; otherwise,
N = ∅.

NN = f([A],[B]).

With all this in hand, validity is defined in the usual way: namely,
as truth-preservation at all normal worlds of all
interpretations.

One could define validity over all worlds (of all interpretations);
however, while this would yield a ‘suitable conditional’
in the going sense, the conditional would be even weaker than the
(very) weak (but nonetheless suitable) conditional generated from the
normal-worlds-only account of validity.

Priest (1992) gives a sound and complete proof theory for the given
semantics, but this is left for the reader to consult.

3.3 A different solution

In very recent work, Field (2008) has championed a non-classical
(paracomplete but non-paraconsistent) theory of truth that involves a
related but, in various ways, more sophisticated approach to a
suitable conditional. In particular, Field gives an extension of the
familiar (Kripkean) Strong Kleene approach to truth, where the
extended language contains a (primitive) suitable
conditional.[6]

3.3.1 A “non-normal” neighborhood framework

For present purposes, I simply sketch (a proper part of) what Field
calls a ‘general semantics’ for his conditional. (More
constraints are needed to give Field's conditional, but the aim here
is simply to sketch the basic idea.) Field also gives an elegant
semantics that utilizes a variation on constructions due to Kripke
(1975) and Brady (1989); however, the ‘worlds’ approach is
sufficient for present purposes.

With the aim of giving an extension of Strong Kleene logic
(see
many-valued logic),
Field proposes the following framework, letting the syntax
be as in the ‘non-normal worlds’ approach above. (Of
course, both approaches are to be extended to the predicate and
quantification cases, but the propositional level gives the basic
idea.) Let W be an infinite set of worlds, with @ a (unique)
“actual” element. In turn, a ‘similarity’ relation is
imposed on W so that each w ∈ W enjoys
a set of ‘similar worlds’ (a so-called
‘neighborhood’ of w). Specifically, Field
proposes that each w ∈ W be assigned a
(possibly empty) directed family Fw that
comprises non-empty subsets of W, where directedness
means

(w ∈ W)(X,Y ∈ Fw)(∃Z ∈ Fw)
Z ⊆ X∩Y

Such directedness of Fw allows for
‘incomparability’, so that the similarity relation needn't
be linear.

Towards avoiding Curry paradox, a few other tweaks are
required. (Towards getting all of the features of Field's conditional,
even more constraints are required, but these are left to Field's work
(2008).) Define the following features for any w ∈
W.

Normality: w is normal just if w ∈ X for all X ∈ Fw.

Non-normality: w is non-normal just if it is not normal.

Loneliness: w is lonely just if {w} ∈ Fw.

Happiness: w is happy just if it is not lonely.

Field's proposal stipulates that @ be both normal and happy on any
interpretation, but otherwise worlds may be non-normal and
lonely. (That @ is to be happy is not merely Field's warmheartedness;
contraction-related validities would emerge were it lonely.)

With this setup, sentences in the conditional-free fragment
are assigned Kleene values (viz., 1, 0.5, 0) at ‘worlds’
in a standard way. In particular, conjunction and disjunction are
min and max, respectively, and negation is 1 minus
the value of the negatum — all relativized to worlds. The given
‘worlds’, of course, play no essential role in any of the
extensional connectives; one needn't ‘look’ at other
worlds to figure out the values of such sentences. The worlds come
into play in the conditional. Where |A|w is the
value of A at w,

|A→B|w = 1 if |A|y ≤ |B|y for some X ∈ Fw and any y ∈ X

|A→B|w = 0 if |A|y > |B|y for some X ∈ Fw and any y ∈ X

|A→B|w = 0.5 otherwise.

With these valuation conditions in place, the validity (or semantic consequence) relation is defined only over ‘actual points’ of all interpretations. Let A be actually verified in an interpretation just if A takes the value 1 at @ in the given interpretation. Then an argument from set S of sentences to conclusion A is valid iff any interpretation that actually verifies S actually verifies A. (Here, a set S of sentences is actually verified just if all of its members are actually verified.)

Two notable features of the conditional

One notable feature of Field's approach is that, with his other constraints (omitted here) in place, the conditional ‘reduces to’ the hook in contexts in which Excluded Middle holds. (See Field 2008 for discussion.) This feature is absent from the simple non-normal worlds approach sketched above.

Another notable feature (in the broader, Fieldian construction) is
that the conditional gives rise to a natural ‘determinacy’
operator — actually, to an infinite hierarchy of stronger and
stronger ‘determinacy’ operators. Given Field's
paracomplete purposes (in which some sentences, like Curry sentences,
are ‘not determinately true’), this is a significant
achievement. For present purposes, however, details are left to
Field's work.

Open Issues and Problems

With either of the foregoing approaches — the simple non-normal or
Field's — one need not reject the existence of Curry sentences (which
are difficult to reject when one's language is a natural language) in
order to keep any of the given ‘naive’ principles (truth,
sets, properties); however, there are various philosophical issues
that need to be addressed, a few of which are canvassed below.

One philosophical issue confronting either approach is the very nature
of such non-normal worlds. What are they? (Of course, they're very
different in the two approaches, but one might ask the question on
either approach.) If, as Field (2008) is inclined to do, one takes a
fairly instrumentalist approach towards the formal semantics, then
perhaps this question is not terribly pressing. If, as Priest is
sometimes inclined to do, one takes a ‘realist’ approach
towards the formal setup, the question seems to be pressing.

Priest's suggestion (1992), with respect to the simple non-normal
framework (or more sophisticated variants), is that non-normal worlds
are simply (impossible) worlds where the laws of logic are
different. But is there any reason, independent of Curry's paradox, to
admit such worlds? Fortunately, the answer seems to be
‘yes’. One reason has to do with the common (natural
language) reasoning involving counter-logicals, including, for
example, sentences such as ‘If intuitionistic logic is correct,
then double negation elimination is invalid’. Invoking
non-normal worlds provides a simple way of modelling such sentences
and the reasoning involving
them.[7]

There are other philosophical (and logical) problems that remain
open. One of the most important recent papers discussing such problems
is Restall 2007. Restall argues that the sorts of non-classical
approach discussed above must give up either transitivity of
entailment, infinitary disjunction or distributive lattice logic
(i.e., an infinitary disjunction operator distributing over finite
conjunction); otherwise, as Restall shows, Curry's paradox arises
immediately and triviality ensues. The importance of Restall's point
lies not only in the formal constraints imposed on suitable
non-classical approaches to Curry; its importance lies especially in
the philosophical awkwardness imposed by such constraints. For
example, one (formal) upshot of Restall's point is that, on a natural
way of modelling propositions (e.g., in familiar world-semantics),
some classes of propositions will not have disjunctions on the (given
sort of) non-classical approach; the philosophical upshot (and
important open problem) is that there is no obvious explanation for
why such classes lack such a disjunction. (Needless to say, it is not
a sufficient explanation to note that the presence of such a
disjunction would otherwise generate triviality via Curry's paradox.)
See Field 2008 for a recent reply to Restall.

The foregoing issues and open problems confront various non-classical
approaches to paradox, problems that arise particularly sharply in the
face of Curry's paradox. It should be understood, however, that such
problems may remain pressing even for those who are firmly committed
to classical approaches to paradox. After all, one might be interested
not so much in accepting or believing such
non-classical proposals, but rather in using such proposals to model
various naive but non-trivial theories — naive truth theory,
naive set theory, semantical property theory, naive denotation theory,
etc.. One need not believe or accept such theories to have an interest
in modeling them accurately. If one has such an interest, then the
foregoing problems arising from Curry's paradox must be
addressed. (See Slaney 1989, and the classic Meyer, Dunn, and Routley
1979, and also Restall 2000 for further discussion.)

Restall, G. 2007. “Curry's Revenge: the costs of non-classical
solutions to the paradoxes of self-reference”, in JC Beall (ed.),
Revenge of the Liar: New Essays on the Paradox, Oxford:
Oxford University Press, pp. 262-271.